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Question
If \[\overrightarrow{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}\] and the coordinates of P are (1, −1, 2), find the coordinates of Q.
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Solution
Given: \[\overrightarrow{PQ} = 3\hat{i} + 2 \hat{j} - \hat{k} .\] Let the position vector of P \[\left( 1, - 1, 2 \right)\] is \[\vec{p}\] such that \[\vec{p} = \hat{i} - \hat{j} + 2 \hat{k}\] and the position vector of \[Q\left( x, y, z \right)\] is \[\vec{q}\] such that \[\vec{q} = x \hat{i} + y \hat{j} + z \hat{k} .\]
Therefore,
\[\therefore \overrightarrow{PQ} = \vec{q} - \vec{p} \]
\[ \Rightarrow 3 \hat{i} + 2 \hat{j} - \hat{k} = \left( x \hat{i} + y \hat{j} + z \hat{k} \right) - \left( \hat{i} - \hat{j} + 2 \hat{k} \right)\]
\[ \Rightarrow 3 \hat{i} + 2 \hat{j} - \hat{k} = \left( x - 1 \right) \hat{i} + \left( y + 1 \right) \hat{j} + \left( z - 2 \right) \hat{k} \]
\[ \Rightarrow x - 1 = 3 , y + 1 = 2, z - 2 = - 1\]
\[ \Rightarrow x = 4 , y = 1, z = 1\] Hence, the coordinates of Q are \[\left( 4, 1, 1 \right)\]
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